Imagine you have a vector say (1, 2)=a right, the length 2 is the portion of (1,2) on the y axis. There I took the dot product a°y^= (0,2) . So the dot product allows you make that same thing with any vector.

I learned it in terms of flux. How many field lines are going through an area, for instance. So let's look at the two extremes.
Remember that an area's vector is pointing out, perpendicular to the length and width of it.
|dw:1345014748903:dw|
So here we see on the left that the cosine of the angle between the two is 0, and the cosine of 0 is 1. So we have 100% flux through the area of electric field lines. Now in the right side we see that the electric field lines are going perpendicular to the area, so none of the field lines are traveling through the area. Cosine of 90 degrees is indeed 0, so we see there is no flux.